Finite Group Databasebeta

← #20273917

Subgroups — #20273917

Order 6784.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811294C211yesmaximal, characteristic
2#16810970C211yesmaximal, characteristic, complemented
3#16810909C211yesmaximal, characteristic, complemented
4#16810828C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810355C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719945—21no—
9C2×#11719923—21no—
10#13133217C2211yescharacteristic
11#13133214C2211yescharacteristic
12#13133213—21no—
13#13133191—21no—
14#13133190—22no—
15C4×Dic106C2211yescharacteristic
16#13133179—21no—
17#13133149—21no—
18C4⋊#88322C2211yescharacteristic
19#13133142—21no—
20#13133140—21no—
21#13133139—22no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×C22⋊D53—22no—
28C2×C22⋊D53—22no—
29#11719945—42no—
30#11719945—42no—
31C2×D212—21no—
32C2×D212—21no—
33C2×D212D412yes—
34C2×C4×D53—22no—
35C2×Dic106—21no—
36C2×Dic106D411yescharacteristic
37C2×C8×C53—21no—
38C2×C8×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C22⋊C4×C53—21no—
41C42×C53C2311yescharacteristic
42#11719923—42no—
43#11719921—22no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53D411yescharacteristic
47#11719919—21no—
48C4×Dic53—21no—
49C2×#88322—21no—
50C23×C53—21no—
51C22×D53—22no—
52D4×C53—21no—
53D4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C22⋊D53—44no—
61C22⋊D53—82no—
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—41no—
65D212—42no—
66D212—41no—
67D212—41no—
68D212—41no—
69D212—41no—
70C4×D53—44no—
71Dic106—21no—
72Dic106—41no—
73Dic106—41no—
74Dic106SD1612yescomplemented
75C8×C53—41no—
76C8×C53—41no—
77#88322—41no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53#12011yescharacteristic
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53C2×SD1611yescharacteristic
88C4×C53#13511yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53AGL(1,ℤ/8)11yescharacteristic
91Dic53—22no—
92Dic53—41no—
93Dic53—81no—
94#1642—531nomaximal, Sylow
95C2×C53—81no—
96C2×C53#44911yescharacteristic
97C2×C53#45011yescharacteristic
98C2×C53#45511yescharacteristic
99D53—82no—
100#550—531no—
101#486—531no—
102#475—531no—
103#461—531no—
104#331—531no—
105Q8⋊C8—531no—
106D4⋊C8D5311yescharacteristic, complemented
107C53#164211yescharacteristic, complemented, Sylow
108C2×#41—1061no—
109C2×SD16—1061no—
110C4⋊Q8—531no—
111#127—531no—
112#127—1061no—
113#121—1062no—
114#121—1061no—
115C4×Q8—531no—
116C4×D4—1061no—
117C4×D4C2×D5311yescharacteristic
118#106—1061no—
119C4⋊C8—531no—
120C4⋊C8C2×D5311yescharacteristic
121#103—1061no—
122#103—1061no—
123#102—1062no—
124C4×C8C2×D5311yescharacteristic
125#41—2122no—
126#41—2122no—
127C2×Q8—531no—
128C2×Q8—1061no—
129C2×D4—532no—
130C2×D4—1062no—
131C2×D4—1062no—
132C2×D4—1061no—
133C2×D4—1061no—
134C2×D4D21211yescharacteristic
135C22×C4—21no—
136C22×C4—1062no—
137SD16—2122no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1061no—
141C4⋊C4—531no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4D21211yescharacteristic
146C22⋊C4—21no—
147C22⋊C4—1062no—
148C42—1061no—
149C42C22×D5311yescharacteristic
150C23—21no—
151C23—1062no—
152Q8—532no—
153Q8—1061no—
154Q8—2121no—
155Q8—2121no—
156D4—21no—
157D4—21no—
158D4—4242no—
159D4—2122no—
160D4—2124no—
161D4—2121no—
162D4—2121no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—1061no—
169C2×C4—1061no—
170C2×C4—2124no—
171C2×C4—2121no—
172C2×C4C2×D21211yescharacteristic
173C2×C4D4×D5311yescharacteristic
174C2×C4D4×D5311yescharacteristic
175C8—41no—
176C8—41no—
177C8—2121no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—2122no—
182C22—2122no—
183C22—2122no—
184C22#1313318711yescharacteristic
185C4—21no—
186C4—41no—
187C4—1062no—
188C4—4241no—
189C4—2121no—
190C4SD16×D5311yescharacteristic
191C4#1313323911yescharacteristic
192C4#1313324111yescharacteristic
193C4#1313324511yescharacteristic
194C2—81no—
195C2—4242no—
196C2#1681077011yescharacteristic
197C2#1681077511yescharacteristic
198C2#1681080211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1642
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #16810970C2semidirect
3. #16810909C2semidirect
4. #16810828C2semidirect
5. #16810366C2semidirect
6. #16810355C2semidirect
7. C53×D4⋊C8C2semidirect
74. Dic106SD16semidirect
106. D4⋊C8D53semidirect
107. C53#1642semidirect